10x = 4.4 (2)

Subtract (1) from (2).

9x = 4 => x = 4/9

(k) Let x = 0.7144. Then

10x = 7.144 (1)
10000x = 7144.144 (2)

Subtract (1) from (2).

9990x = 7137 => x = 7137/9990 = 793/1110

PROBLEM

Express each of the following decimal numbers as a binary number, accurate to twelve places to the right of the decimal point.
(a) 2 1/4
(b) 4 3/32
(c) 0.1
(d) 3/4
(e) 3.61

SOLUTION

The general procedure for converting a decimal number into a binary number is to write each decimal number as a linear combination of powers of 2, and then write the result as a binary (base two) number.

(a) 2 1/4 = 2 + 1/4 = 10.012

(b) 4 3/32 = 4 + 1/16 + 1/32 = 100.000112

(c) 0.1 = 1/10

The largest power of two which is less than 1/10 is 1/16. So

0.1 = 1/16 + x

where

x = 1/10 - 1/16 = 8/80 - 5/80 = 3/80 => 0.1 = 1/16 + 3/80

Since 3/80 is equal to 1/26, the largest power of two which is less than 3/80 is 1/32. So

The largest power of two which is less than 1/1600 is 1/2048. So

3.61 = 2 + 1 + 1/2 + 1/16 + 1/32 + 1/64 + 1/2048 + x

where

Since 7/51200 = 1/7314.285..., the largest power of two which is less than 7/51200 is 1/8192. So

3.61 = 2 + 1 + 1/2 + 1/16 + 1/32 + 1/64 + 1/2048 + 1/8192 + x

where

x = 7/51200 - 1/8192 = 28/204800 - 25/204800 = 3/204800
=> 3.61 = 2 + 1 + 1/2 + 1/16 + 1/32 + 1/64 + 1/2048 + 1/8192 + 3/204800 = 11.1001110000102
Note: The lowest common denominator of two fractions is obtained by factoring the denominators of each of the fractions and calculating the product of the highest powers of each prime factor present in either of the denominators. For example, to obtain the lowest common denominator of 1/1600 and 1/2048, we first factor 1600 and 2048. We have

Determine how many times 130-, where "-" is some digit between zero and nine, will go into 9600 without exceeding it. Write this number, 7, at the top, with a decimal point to the left of it, and where the "-" was.

25x + 10y = 775 (1)
x + y = 40 => y = 40 - x (2)

Substituting (2) into (1),

Tommy has 25 quarters and 15 dimes.

PROBLEM

Jack can shovel the snow on someone's property in 5 hr. Joe can shovel the snow in 3 hr.
(a) How long will it take Jack and Joe, working together, to shovel the snow?
(b) What fraction of the entire job does each one do?
(c) Assume that each is paid at the same hourly rate they would get if they did the job alone and that the total each would get paid if they worked alone is $75. Who gets paid more if they work together?

SOLUTION

(a) The fraction of the job that (Jack, Joe) does per hour if working along is (1/5, 1/3). Let t be the time (in hours) it takes for both of them to do the job together. Then

(1/5 + 1/3)t = 1

Multiply this equation by 15.

(3 + 5)t = 15 => 8t = 15 => t = 15/8 hr = 1.875 hr

(b) The fraction of the job that (Jack, Joe) does is [(1/5)(15/8), (1/3)(15/8)] = (3/8, 5/8).

PROBLEM

A ranger in an observation tower can sight the north end of a lake 15 km away and the south end of the same lake 19 km away. The angle between these two lines of sight is 104&deg. How long is the lake?

where c0, c1,..., and cn are constants, and suppose that we want to divide this polynomial by the binomial x - a, where a is also a constant. We may proceed as in ordinary long division by writing

While in ordinary long division we work first with the most significant digits in the dividend and work our way towards the less significant digits, in dividing the above polynomial by the binomial, we start with the term which is of highest order in x and work our way towards the lower-order terms.

The highest-order term in the quotient will necessarily be cnxn-1 since, when this is multiplied by x - a, the result includes the term cnxn. Thus, we write cnxn-1 above the horizontal line.

The constants cn, dn-1, dn-2,..., dk,..., d2, and d1 that appear on the bottom line are the coefficients of xn, xn-1, and x in (1), while d0 is the remainder. Thus, the procedure just described is a faster way of obtaining the quotient in (1).

PROBLEM

Consider three opaque boxes. One box contains all white balls, one all black balls, and one a mix of black and white balls. Each box is labeled, but the labels are all wrong. How many balls would you need to pull out to determine which box is which?

SOLUTION

Suppose the boxes are labeled as shown. Then, because we are told that the labels are all wrong, the possible types of balls in each box are as written below the drawing of the box (B = black, W = white, B/W = black and white).

Thus, box 1 actually contains either all white or mixed black and white balls, box 2 actually contains either all black or mixed black and white balls, and box 3 actually contains either all black or all white balls.

Suppose we draw one ball from box 3. If the ball is black, this means that box 3 contains all black balls, box 2 contains mixed black and white balls, and box 1 contains all white balls. If the ball drawn from box 3 is white, this means that box 3 contains all white balls, box 1 contains mixed black and white balls, and box 2 contains all black balls. Thus, by picking one ball from box 3, it is possible to determine what kind of balls is contained in all three of the boxes.

PROBLEM

A car's odometer read 14632 miles on 26 October 2007, 16686 miles on 8 March 2008, and 20972 miles on 28 December 2008. Estimate the number of miles driven from 28 December 2007 to 28 December 2008 assuming that the miles driven each day is the same.

SOLUTION

From 26 October 2007 to 28 December 2007 is 63 days. 2008 is a leap year, so there are 29 days in February 2008. From 26 October 2007 to 8 March 2008 is therefore 134 days. On 28 December 2007, the odometer should have read about 14632 miles + (16686 miles - 14632 miles)(63 days) / (134 days) = 15597.69 miles. The number of miles driven from 28 December 2007 to 28 December 2008 was about 20972 miles - 15597.69 miles = 5374.313 miles => 5374 miles.

PROBLEM

John has two $44 vouchers which can be used to pay for monthly subway passes. The cost of a subway pass increases from $44 to $59 next month. John has the option of purchasing additional $21, $31, $35, $44, or $50 vouchers which are deducted, pretax, from his paycheck. John is in the 21% income tax bracket; 21% of his paycheck is deducted for income tax. What additional vouchers should John purchase in order to buy two subway passes next month? Assume that no change is given if the voucher value exceeds the cost of the subway passes.

SOLUTION

John has $88 of vouchers. The total cost of two subway passes next month is (2)($59) = $118, so John needs $118 - $88 = $30 more. He can either buy one $21 voucher and pay $9 aftertax cash or buy a $31 voucher and forfeit $1 since no change is given.

If he buys the $21 voucher and pays $9 aftertax cash, his total additional pretax cost is $21 + $9/(1 - 0.21) = $32.39. If he buys the $31 voucher his total additional pretax cost is $31. So John is better off buying the $31 voucher and forfeiting $1.

PROBLEM

Gasoline is purchased for a car on the dates shown in the table below. Each time gas is purchased, the tank is filled up. The table shows the mileage on the car's odometer, the price paid, and the number of gallons of gasoline purchased.

Date

Odometer Reading (miles)

Gasoline Purchased (gallons)

Total Cost

Cost per Gallon

2007 Apr 1

11640

9.553

$25.40

$2.659

2007 Apr 13

11778

12.983

$35.69

$2.749

2007 May 12

12040

15.255

$44.84

$2.939

From the data in the table, determine the miles per gallon and the cost per mile achieved by the car.

SOLUTION

From the data, the total number of miles driven during the period shown is 12040 miles - 11640 miles = 400 miles. The total gas used is 12.983 gal + 15.255 gal = 28.238 gal. The total cost of the gas used is $35.69 + $44.84 = $80.53. The miles per gallon achieved by the car is (400 miles) / (28.238 gal) = 14.17 mpg. The cost per mile is $80.53 / (400 miles) = $0.2013 per mile. Note that we neglected the initial purchase of gas in our calculations because it was used to fill up the tank prior to the period during which the 400 miles were driven.
Back to Main Menu

PROBLEM

Jack, a taxpayer, is in the 35% income tax bracket.

(a) If his annual income in 2007 was $75,000, and his income tax was deducted from his paycheck throughout the year at the rate corresponding to his tax bracket, how much income tax did he pay in 2007?

(b) Jack paid $20,000 in property tax in 2007. If this amount is deductible for income tax purposes (i.e., it counts as negative income), how much of an income tax refund should he get when he files his income tax forms?

(c) If the $20,000 in property tax is not deductible for income tax purposes but is taxed at 28% instead of 35%, how much of an income tax refund should he get?

SOLUTION

(a) Jack paid (0.35)($75,000) = $26,250 income tax in 2007.

(b) The income tax that Jack should pay for 2007 is (0.35)($75,000 - $20,000) = $19,250. Since $26,250 was deducted from his paycheck, he should get a $26,250 - $19,250 = $7,000 refund.

(c) The income tax that Jack should pay for 2007 is (0.35)($55,000) + (0.28)($20,000) = $24,850. Since $26,250 was deducted from his paycheck, he should get a $26,250 - $24,850 = $1,400 refund.

PROBLEM

Joan, a taxpayer, earned $85,000 in 2007.

(a) Joan's income tax was deducted from her paycheck throughout the year at the rate corresponding to her tax bracket. If Joan is in the 35% tax bracket, how much income tax was deducted from her paycheck?

(b) Joan paid $10,000 in property tax in 2007. If this amount is taxed at 28% instead of 35%, how much of a refund should Joan get when she files her income tax forms?

(c) How would your answer to part (b) change if Joan was in the 25% tax bracket?

SOLUTION

(a) (0.35)($85,000) = $29,750 was deducted from Joan's paycheck.

(b) Joan should pay (0.35)($75,000) + (0.28)($10,000) =
$29,050 income tax for 2007, so she should get a refund of $29,750 - $29,050 = $700.

(c) If Joan was in the 25% tax bracket, (0.25)($85,000) = $21,250 would have been deducted from her paycheck. The amount of income tax she should actually pay is (0.25)($75,000) + (0.28)($10,000) = $21,550. So she would owe $21,550 - $21,250 = $300 in additional income tax when she files her income tax forms.